A selection procedure for extracting the unique Feller weak solution of degenerate diffusions

TL;DR

This paper establishes a method to select a unique Feller weak solution for certain degenerate diffusions by analyzing the small noise limit of non-degenerate approximations, using viscosity solutions and comparison principles.

Contribution

It extends previous work by providing a selection procedure for the unique Feller solution in degenerate diffusions through viscosity solution techniques.

Findings

Proves the existence of a unique Feller limit for degenerate diffusions.

Establishes a comparison principle for viscosity solutions.

Demonstrates a correspondence between Feller solutions and viscosity solutions.

Abstract

In this work, we show that for the martingale problem for a class of degenerate diffusions with bounded continuous drift and diffusion coefficients, the small noise limit of non-degenerate approximations leads to a unique Feller limit. The proof uses the theory of viscosity solutions applied to the associated backward Kolmogorov equations. Under appropriate conditions on drift and diffusion coefficients, we will establish a comparison principle and a one-one correspondence between Feller solutions to the martingale problem and continuous viscosity solutions of the associated Kolmogorov equation. This work can be considered as an extension to the work of V. S. Borkar and K. S. Kumar (2010).